m o n y for the Twenty-Fif th Internat ional Congress of Mathematicians.

I 'm sure each indiv idual r eader w o u l d find m a n y things of interest and m a n y new ideas in Poincar~'s Prize, I will just men t i on a few that came my way, in o r d e r to suggest that there is i ndeed a g e n u i n e r ichness that lies be- t w e e n the covers of this book .

Much to m y a m a z e m e n t I l ea rned that the Poincare d o d e c a h e d r a l space, which he d i s cove red in 1904 as a coun- t e r example to the " theorem" he had c la imed in 1900, to this day remains the on ly k n o w n c o u n t e r e x a m p l e to this false theorem. I l ea rned that RH were not the initials of that giant of topol- ogy, R H Bing, but we re in fact his ac- tual first name. More impor tant ly , I also lea rned that Bing, w h o h imse l f made p ro longed ser ious a t tempts on the Poincare Conjecture, in the end may have conc luded that the conjec ture was false. This, for me, was one of the very bes t momen t s in Szpiro 's b o o k because it s h o w e d so dramat ica l ly h o w hard it is in mathemat ics to k n o w wha t is true and what is false if s o m e o n e of R H Bing's s tature can be so comple t e ly w r o n g abou t such a fundamen ta l ques- t ion as the Poincare Conjecture.

From a more pe r sona l po in t of view, I was de l igh ted to learn that James Alexander , w h o is well k n o w n to most mathemat ic ians as the d i scovere r of the famous and fabulous "Alexander ho rned sphere ," has a classic technical cl imbing route up 14,255 foot-high Longs Peak in Colorado , n a m e d Alexander ' s Chimney, after him. Simi- larly, I f o u n d a ra ther l eng thy history of the t~cole P o l y t e c h n i q u e - - w h i c h Poincare a t t e n d e d from 1873 to 1875, graduat ing s e c o n d in his c l a s s - - a b - solutely fascinat ing because m y wife di- rec ted a p rog ram at Co lo rado College for several summers dur ing the mid- 1980s. That p r o g r a m was d e s i g n e d for s tudents from the l~cole Po ly techn ique to guide t hem in the s tudy of English and to acqua in t t hem with American culture.

Here is one final nugge t in the same vein that takes on spec ia l i rony n o w that Pere lman has re jec ted the Fields Medal: at his dea th in 1912, Poincare had rece ived the largest n u m b e r of nomina t ions for a Nobe l Prize of any non-winner . Recall that there is not a mathemat ics ca tegory for Nobe l Prizes

and that it is for this r ea son that the Fields Medal is c o n s i d e r e d to be the mathematical equ iva len t of a Nobel Prize. Poincare 's nomina t ions were all in physics.

I must admit that, in the end , Szpiro does deliver on the t ab lo id p romise from the back cove r of his b o o k . The most gripping, h a r d - t o - p u t - d o w n read- ing are Szpiro's last two chapters , w h e n he finally gets d o w n to d i scuss ing the very messy con t roversy su r round ing the solution of Po inca re ' s f amous con- jecture. It is ha rd not to b e in t r igued by this controversy. There are some very serious i ssues here: Wha t share of the credit for the so lu t ion d o e s Hamil- ton deserve? That the Clay Inst i tute may wel l award him a large share o f the mil- l ion-dol lar prize is just one me a su re of the fact that m a n y p e o p l e be l i eve that he and Pere lman share equa l ly in ar- riving at the final solut ion. Ano the r of the serious issues is the w a y in wh ich Perehnan b y p a s s e d t radi t ional ly ac- cep ted methods for pub l i sh ing mathe- matical proofs b y p lac ing his unrefer- eed proofs on the Internet . It has taken three years and severa l t eams of heav- ily f inanced exper t s to c o n c l u d e that Perelman 's work is correct . Meanwhi le , the really messy par t of the con t roversy arose from a c la im m a d e by a team of Chinese mathemat ic ians that they had publ i shed the first c o m p l e t e p r o o f of the Poincare Conjecture. The article in The Neat' Yorker great ly in f lamed this controversy by inc lud ing a ful l -page drawing with Shing-Tung Yau look ing as if he is about to rip the Fie lds Medal from the neck o f Pe rehnan ( w h o looks a bit like Vincent Van G o g h in this drawing).

Szpiro sorts t h rough the detai ls and complex i t i e s - - i nc lud ing the ethical o n e s - - o f this con t rove r sy qui te thor- oughly and with wha t s e e m s to be con- s iderable fairness and a g rea t dea l of wisdom, both c onc e rn ing h u m a n na- ture and with r e spec t to ma in ta in ing al- ways the highest r ega rd for the wel l- be ing of mathemat ics . Thus, he is ab le to br ing us b e y o n d the con t rove r sy to the poin t whe re w e can ce l eb ra t e the solut ion of the Po incare Conjecture, pe rhaps dream of so lu t ions to one of the six remaining mi l l enn ium prob- lems, and f ind o the r w a y s - - i n the words of the miss ion s t a t emen t of the Clay Ins t i tu te - -" to fur ther the beauty,

power , and universal i ty of ma themat i - cal thinking."

R E F E R E N C E S

1. N. L. Biggs, E. K. Lloyd, and R. J. Wilson,

Graph Theory 1736-1936, Clarendon

Press, 1976. 2. P. Hoffman, The Man Who Loved Only Num-

bers: The Story of Paul Erdds and the Search

for Mathematical Truth, Hyperion, 1998.

3. D. Mackenzie, The Poincare Conjecture--

Proved, Science 314 (22 December 2006),

1848-1849. 4. S. Nasar and D. Gruber, Manifold Destiny,

The New Yorker (August 28, 2006), 44-57.

5. J. J. Watkins, Across the Board: The Math-

ematics of Chessboard Problems, Prince-

ton University Press, 2004.

More math comics by Cour tney G i b b o n s are ava i lab le on l ine at: b r o w n s h a r p i e . c o u r m e y g i b b o n s . o r g

Department of Mathematics and Computer Science

Colorado College Colorado Springs, CO 80903 USA e-maih jwatkins@coloradocollege.edu

Fixing Frege John Burgess

PRINCETON AND OXFORD: PRINCETON UNIVERSITY

PRESS, 2005. PP. xii + 257. ISBN 0-691-12231-8,

US$ 39 .95

Reason's Proper Study: Essays Towards a Neo- Fregean Philosophy of Mathematics Bob Hale and Crispin Wright

OXFORD, CLARENDON PRESS, 2001. PP. xiv + 455.

US$ 45 .00 ISBN 0-19-823639-5

REVIEWED BY ~YSTEIN LINNEBO

W e k n o w that there are infi- ni tely m a n y p r ime number s and that every natura l n u m b e r

�9 2007 Springer Science+Business Media, Inc., Volume 29, Number 4, 2007

has a un ique p r ime factorization. What sort of k n o w l e d g e is this? Unlike our k n o w l e d g e that Mogad ishu is the capi- tal of Somal ia or that e lec t rons have negat ive charge, ar i thmetical knowl- edge does not s eem to be empirical ; that is, it d o e s no t s e e m to be b a s e d on observa t ion or exper iment . The Ge rman mathemat ic ian , logician, and phi loso- phe r Got t lob Frege (1848-1925) devel- o p e d a b o l d n e w account of the na ture of ar i thmetical knowledge : He a rgued thatpure logic prov ides a source of such knowledge , and that ar i thmetic there- fore is a priori rather than empirical . This v iew is n o w k n o w n as logicism and is one of the ma in ph i losoph ica l accounts of mathemat ics (a longs ide for- malism, intuit ionism, convent ional ism, and structuralism).

Frege 's de fense of his logicist v iew of ar i thmetic p r o c e e d s in two steps. The first s tep consis ts in an account of h o w number s are a p p l i e d and of their iden- tity condi t ions . Frege argues that count- ing involves the ascr ip t ion of number s to concepts . For instance, w h e n w e say that there are e ight planets , w e ascr ibe the n u m b e r e ight to the concep t " . . . is a planet". Let '#' abbrev ia t e the op- era tor ' the n u m b e r of'. Frege ' s claim is then that '#' app l i e s to any concep t F to form the exp re s s ion '#F', meaning "the n u m b e r of Fs". Next Frege argues that the n u m b e r of Fs is ident ical to the n u m b e r of Gs if a n d on ly if the Fs and the Gs can b e pu t in a o n e - t o - o n e cor- r e s pondence . This pr inc ip le (which is typical ly assoc ia ted with Georg Cantor) is k n o w n in the ph i losoph ica l l i terature as Hume's Principle (s ince it may have b e e n an t i c ipa ted by the ph i l o sophe r David Hume) . In o rde r to formal ize this pr inciple , Frege m a k e s essent ia l use o f the fact that his logic is second-order ; that is, in add i t i on to the o rd ina ry first- o rde r quantif iers V x a n d 3x, wh ich range ove r s o m e d o m a i n D, Frege 's logic also has s e c o n d - o r d e r quantif iers VR and 3R, w h i c h range over relat ions o n D (of s o m e par t icular adicity). Let ' F ~ G ' abbrev ia t e the pure second- o rde r s ta tement that there is a relat ion R that one - to -one corre la tes the F s and the Gs. H u m e ' s Pr inciple can then be exp re s sed as:

(HP) #F = #G<--+ F -~ G

This m a k e s s ense b e c a u s e ~ is an equ iva l ence relat ion.

The s e c o n d step of Frege 's defense of logic ism provides an explicit defini- t ion o f terms of the form '#F'. Frege d o e s this in a theory that consists of s e c o n d - o r d e r logic and his "Basic Law V," which states that the extens ion of a concep t F is identical to that of a con- cep t G if and only if the Fs and the Gs are co-extensional ; or, in con tempora ry nota t ion

(V) {x IFx } = {x]Gx} e--, V x ( F x ~ Gx).

In this theory , Frege defines # F as the ex t ens ion of the concept "x is an ex- t ens ion of s o m e concept equ inumerous wi th F." That is, he def ines

# F = I x l3G(x = {ylGy} /~ F-~ G)}.

This defini t ion is easily seen to sat- isfy (HP). More interestingly, Frege p roves in meticulous technical detail h o w this definit ion and his theory of ex- tensions entail all of ordinary arithmetic.

Howeve r , just as the second vo lume of his magnum opus was going to press in 1902, Frege received a letter from the English logician and ph i losopher Ber t rand Russell, who repor ted that he had "encoun te red a difficulty" with Frege ' s theory of extensions. The diffi- culty Russell had encoun te red is the p a r a d o x n o w bearing his name. Frege 's t heo ry of extens ions is in effect a naive t heo ry of sets. We may thus cons ider the set of all sets that are not member s of themselves . In Frege's theory w e can then p r o v e that this set bo th is and is no t an e l e m e n t of itself. Frege 's re- s p o n s e to Russell 's letter is remarkable . Sixty years later Russell descr ibed it as follows.

As I th ink about acts of integrity and grace, I real ise that there is noth ing in m y k n o w l e d g e to compare wi th Frege ' s ded ica t ion to truth. His en- tire l ife 's w o r k was on the verge of comple t i on , much of his work had b e e n i gno re d to the benef i t of men inf ini tely less capable , his s econd v o l u m e was about to be publ i shed , a n d u p o n finding that his funda- menta l a s sumpt ion was in error, he r e s p o n d e d with intellectual plea- sure, c lear ly submerging any feel- ings of pe rsona l d isappoin tment . Russell 's pa radox eventual ly led

Frege to give up on logicism. Until the 1980s bo th logicians and ph i losophers r e g a r d e d F regean logicism as a dead

end, and p e o p l e at t racted to the idea of logic ism p u r s u e d other ve r s ions of it, such as Russell 's very c o m p l i c a t e d "type theory."

However , over the past two decades there has been a resurgence of interest in Fregean logicism. A variety o f con- sistent f ragments of Frege 's t heo ry have b e e n ident if ied and explored , a n d their poss ib le phi losophica l s ignif icance has b e e n v igorous ly debated . The two b o o k s under rev iew are wi thout doub t a m o n g the most important p roduc t s of this resurgence. Reason's Proper Study is the most extens ive ph i losoph ica l ar- t iculat ion and defense to da te o f a spe- cific neo-Fregean programme, whe rea s Fixing Frege offers the d e e p e s t and most comprehens ive technical investi- ga t ion of a variety of different neo- Fregean approaches .

Neo-Fregean i sm began with Crispin Wright (Frege's Conception of Numbers as Objects, 1983) w h o sugges ted that the p r o b l e m p o s e d by Russell 's p a r a d o x be e v a d e d by mak ing do with the first s tep of Frege 's approach , a b a n d o n i n g alto- ge the r the s e c o n d step and its incon- sistent theory of extensions. This ap- p roach is m a d e poss ib le by two relatively recent technical discoveries. The first d i scovery is that (HP), unl ike (V), is consistent . More precisely, let Frege Arithmetic be the s econd-o rde r theory, with (HP) as its sole non- logical axiom. Frege Arithmetic can then b e shown to be consistent if and only if s e c ond -o rde r Peano Arithmetic is. The s e c o n d d iscovery is that Frege Arith- metic and some very natural defini t ions suffice to der ive all the ax ioms of sec- ond -o rde r Peano Arithmetic. This result is k n o w n as Frege's Theorem. It is an amaz ing result. For more than a century now, informal ari thmetic has almost wi thout excep t ion been given some P e a n o - D e d e k i n d style axiomatizat ion, w h e r e the natural numbers are r ega rded as finite ordinals , def ined by their po- sit ion in an omega-sequence . Frege 's T h e o r e m shows that an al ternat ive and concep tua l ly comple te ly different ax- iomat iza t ion of arithmetic is possible , b a s e d on the idea that the natural num- bers are finite cardinals, def ined by the cardinal i t ies of the concepts whose numbe r s they are.

Technically speaking, the neo- F regean founda t ion of ari thmetic is thus a success: it is consis tent and strong

8 4 THE MATHEMATICAL INTELLIGENCER

enough to p r o v e all of o rd inary arith- metic. But wha t abou t its philosophical significance?

Reason's Proper Study, which brings together 15 essays by the two foremost neo-Fregeans, is an ex tended argument that neo-Fregeanism is a phi losophical success as well. It is a rgued that this ap- proach enjoys most of the phi losophical benefits p romised by Frege's original but sadly inconsistent form of logicism. I can here ment ion only three of the questions that Hale and Wright g rapple with in their defense of this claim.

The first ques t ion is w h e t h e r the first of the two s teps of Frege 's app roach (which I d e s c r i b e d above) can stand on its own. Frege h imsel f thought it could not because (HP) fails to settle all ques- tions about the ident i ty of numbers . For instance, (HP) fails to settle whe the r the number of p lane t s is identical to the Ro- man e m p e r o r Jul ius Caesar! In order to settle such p e s k y quest ions, Frege thought it necessa ry to p r o c e e d to the second s tep and give an explici t defin- ition of the natural numbers .

Hale and Wright disagree, arguing in- s tead that (HP) gives the criterion of identity for numbers ; that non-numbers have different such criteria; and that this implies that Caesar cannot be a number.

The s e c o n d quest ion concerns the epis temological status of Hume 's Princi- ple. As Hale and Wright admit, (HP) does not par t icular ly "look like" a logi- cal principle. They de fend instead the slightly w e a k e r claim that (HP) can serve as an exp lana t ion of the mean ing of the #-opera tor and thus be k n o w n a priori. If correct, this c la im w o u l d be very sig- nificant, as it w o u l d establ ish that arith- m e t i c - w i t h its infinite on to logy of num- bers----can be k n o w n a priori. This wou ld be a lmos t as g o o d as what was promised by Frege 's original logicism.

The third ques t ion concerns the de- mand for a d e e p e r and more general unders tand ing of the k ind of explana- t ion that (HP) is s u p p o s e d to provide. The d e m a n d is m a d e part icularly acute by the structural similarity be tween (HP) and the inconsis tent pr inciple (V). What if it is just a h a p p y acc ident that (HP) is consistent? If so, the neo-logi- cists can ha rd ly claim that mere ly lay- ing (HP) d o w n as an exp lana t ion of the mean ing of the # -ope ra to r yields a p r i - ori k n o w l e d g e that (HP) is true. For surely a be l ie f canno t count as knowl-

edge if it is just a h a p p y acc iden t that it is tree! Hale and Wright r e s p o n d by arguing that k n o w l e d g e does not re- quire any kind of an teceden t guaran tee against error. It seems to this rev iewer that this can at mos t pos tpone , not elim- inate, the need for a d e e p e r exp lana- tion. After all, it is part of the very na- ture of bo th mathemat ics and ph i lo sophy to seek genera l explana- tions wheneve r such are possible .

Whereas the agenda of Reason's Proper Study is p redominan t ly philo- sophical , that of Fixing Frege is pre- dominant ly mathematical . For instance, Fixing Frege has little to say abou t the first two quest ions men t ioned a b o v e but a lot to say abou t the third.

The open ing chap te r p rov ides a very readable in t roduct ion to the mathemat- ical aspects of neo-Fregeanism. Burgess first provides a useful summary of Frege 's own construct ions, of Russell 's pa r adox and Frege ' s r e sponse to it, and of Russell 's compe t ing form of logicism. He then deve lops a sophis t ica ted frame- work in which var ious neo -F regean the- ories can b e ana lyzed and their strengths compared . Particularly useful is his exp lana t ion of a h ie rarchy of mathematical theories, ranging from very weak subsys tems of ar i thmetic up to very s trong systems of set theory. This hierarchy provides a unif ied sys- tem of targets for neo -F regean recon- struction, which enables us to measure the strength of a neo -F regean theory in terms of how much of this h ie rarchy the theory al lows us to reconstruct .

The remaining two chapters explore the two main ways of ensur ing the con- sistency of Frege- inspired theories. The standard w a y - - a l r e a d y encounte red above and the topic of Chapter 3 - - i s to abandon Frege's Basic Law V in favor of related but w e a k e r principles such as (HP). An al temative w a y - - w h i c h forms the topic of Chapter 2 - - i s by placing re- strictions on which open formulas can define relations. In order to expla in this alternative way w e need some defini- tions. A comprehens ion ax iom is an ax- iom which states that an open formula 4~, with free variables xl, . . . , Xn, suc- ceeds in defining an n-adic relation R, under which n objects fall if and only if they satisfy the formula 4~, or in symbols:

3RVxa . . . V X n [ R X l , . �9 �9 , X n 4-'+

C/)(Xl . . . . . Xn)]

A c o m p r e h e n s i o n ax iom is sa id to be predicative if 4~ conta ins n o s e c o n d - o r d e r quantif iers and impredicative otherwise . If r ega rded as def in i t ions , p red ica t ive c o m p r e h e n s i o n ax ioms have a nice p rope r ty of non-c i rcu lar i ty , n a m e l y that the re la t ion R is de f i ned wi thou t quant i fying over a to ta l i ty that inc ludes R itself.

The ph i lo sophe r Michael D u m m e t t has p r o p o s e d an excit ing bu t cont ro- versial analysis of "the cause" of Rus- sell 's paradox: He b lames the cont ra- d ic t ion not on Basic Law V bu t ra ther on the p resence of impred ica t ive com- p r e h e n s i o n axioms in the b a c k g r o u n d theory. To substant iate this analysis , it must be shown that restr ict ing onese l f to predica t ive c o m p r e h e n s i o n res tores consis tency. Chapter 2 gives a n ice pre- senta t ion of some earl ier t h e o r e m s w h i c h show that this is i n d e e d the case. But for the analysis to be p laus ib le , it mus t also be s h o w n that this res t r ic t ion leaves the charac ter of the r e l evan t the- or ies intact; o therwise cons i s t ency will be res tored not by excis ing a p rec i se ly c i rcumscr ibable "cause of p a r a d o x " but m o r e blunt ly by render ing the theor ies impotent . But some n e w resul ts f rom Chap te r 2 show that the resu l t ing the- or ies are very weak . So this b o d e s ill for Dummet t ' s claim that impred ica t ive c o m p r e h e n s i o n is "the s e rpen t that en- t e red Frege 's paradise."

The final chap te r examines the stan- da rd way of restoring cons is tency . Call a pr inc ip le of the logical form

(*) ~ F = ~ G ~ F - - G

an abstraction principle. Some abs t rac- t ion p r i n c i p l e s - - s u c h as ( H P ) - - a p p e a r to be accep tab le , w he re a s o t h e r s - - s u c h as ( V ) - - c l e a r l y are not. Can a sha rp a n d we l l -mot iva t ed line be d r a w n b e t w e e n abs t rac t ion pr inc ip les that a re accep t - ab le and those that are not? A na tura l t hough t is that (V) is m a d e u n a c c e p t - ab le by the fact that it r equ i r e s a one - t o - o n e m a p from the c o n c e p t s o n a do- ma in into the d o m a i n itself, w h i c h w e k n o w by Cantor ' s t h e o r e m to be im- poss ib le . Say that an abs t r ac t ion pr in- c ip le (*) is non-inf la t ionary o n a do- ma in D if the equ iva lence re la t ion -- is such that there are no m o r e - - - equ iva - l ence classes of concep t s than the re are ob jec t s in D. O n e easi ly sees that eve ry non- inf la t ionary abs t rac t ion pr inc ip le has a mode l .

�9 2007 Springer Science+Business Media, Inc., Volume 29, Number 4, 2007 8 5

The ques t ion of w h e n a system of abst ract ion pr inciples is accep tab le is harder . For even if the pr inc ip les that m a k e up the system are individual ly non-inf lat ionary, the sys tem as a who le n e e d not be. This ques t ion receives a pene t ra t ing analysis in Kit Fine 's The Limits of Abstraction (2002), w h e r e it is s h o w n that a sys tem of abs t rac t ion prin- c iples is accep tab le p r o v i d e d that each pr inciple is individual ly non- inf la t ionary and based on an equ iva lence relat ion that satisfies a certain "logicali ty con- straint" ( involving invar iance unde r per- muta t ions of the d o m a i n D). In Fixing Frege, Burgess gives a n ice expos i t ion of this analysis and substant ia l ly ad- vances the discuss ion by p inpo in t ing the s t rength of the resul t ing system: that of "third-order arithmetic."

A l t h o u g h the s t r eng th o f this sys- t e m is thus subs tan t ia l , it falls far shor t o f the s t r o n g e r ta rge t t h e o r i e s in Burgess ' s h ie ra rchy . In o r d e r to r each h igher , Burgess p r o p o s e s a n e w w a y to mot iva t e the ax ioms o f o r d i n a r y set theory . He first uses " l imi ta t ion of s ize" c o n s i d e r a t i o n s to m o t i v a t e a so- ca l l ed re f lec t ion p r inc ip le , f rom w h i c h he de r ives ( d r a w i n g on ea r l i e r w o r k b y Paul Be rnays ) mos t o f the a x i o m s o f ZFC set theory , as we l l as s o m e la rge ca rd ina l h y p o t h e s e s . A l t h o u g h this is an impre s s ive feat , Burges s ad- mi ts that the m o t i v a t i o n a n d the re- su i t ing t h e o r y are no l o n g e r par t i cu- lar ly Fregean .

S u m m i n g up , it is n o w clear , in a w a y it was no t two d e c a d e s ago, that a wea l t h of p h i l o s o p h i c a l a n d techni - cal insights can b e r e s c u e d f rom the ru ins of Frege ' s logic ism. W h e t h e r t h e s e insights a d d up to a c o h e r e n t a n d at t ract ive p h i l o s o p h y o f ma the - mat ics is still ( in m y o p i n i o n ) an o p e n ques t ion . But Reason's Proper Study a n d Fixing Frege are w a r m l y r ecom- m e n d e d as the bes t p l a c e s to start for an e x a m i n a t i o n of, r e spec t ive ly , the p h i l o s o p h i c a l and t echn ica l ins ights to b e lea rn t f rom Frege a n d the p r o s p e c t s for a n e o - F r e g e a n p h i l o s o p h y of math- emat ics .

Department of Philosophy University of Bristol 9 Woodland Rd Bristol BS8 1TB UK e-maih oystein.linnebo@bris.ac, uk

Letters to a Young Mathematician by/an Stewart

NEW YORK, BASIC BOOKS, 2006. HARDCOVER

US $22.95 ISBN: 9780465082315

REVIEWED BY REUBEN HERSH

Dear grandchi ldren David, Jessica, and Ze'ev,

As is cus tomary, I list you in chrono- logical order , by your year of birth. As all of you p r o b a b l y know, your grand- father in New Mexico is a ret ired math professor . If it should some day hap- pen , by some g o o d b o o k or some good teacher, that one or two of you get tu rned on to math, I'll be more than ea- ger to he lp you, with my information and my advice. In all fairness then, I am lett ing you know that the new b o o k by my fr iend Ian Stewart a l ready offers such informat ion and advice. Therefore, for you three, and also for any other potent ia l r eaders of this prest igious pe- riodical, I am going to tell you about Stewart 's book .

First I'll tell you about his informa- tion. Then I'll tell you about his advice. And finally, I'll tell you my o w n advice.

The b o o k consists of 21 letters to "Meg." The quota t ion marks inform us, I suppose , that "Meg" is imaginary or fictitious. In the first letter "Meg" is "at school." Since Ian doesn ' t hesitate to speak ser iously and deep ly to "Meg", I guess she 's a l ready in wha t we here in the States call "high-school." Meg is wonde r ing wha t mathemat ic ians do, and h o w her Uncle (??) Ian became a mathemat ic ian . As time passes, Meg chooses to s tudy math(s) at University. By the last letter, she's conce rned with wha t a tenure- t rack Assistant Professor is conce rned with: the mad struggle to attain tenure. ("Tenure" is the col lege professors ' w o r d for "job security.)

I am sure you, or anyone interested in mathemat ica l life today, will find the "Letters" interest ing and enjoyable. Stewart freely confides to Meg some of his own pe r sona l s tow, of h o w he was d rawn to mathemat ics , and of some of his p leasures a n d successes as a math- ematician. There is a really wonderfu l account of h o w an investigation into the abstract theory of groups turned out to

be of great use in analyzing the gait of four - footed creatures, like dogs! Who k n e w that there was even such an aca- demic specia l ty as "Gait Studies"?

Several o f the chapter titles tell e n o u g h to m a k e clear their messages: "The Breadth of Mathematics," "Hasn't it All Been Done? . . . . How Mathemati- cians Think," "How to Learn Math," "Fear of Proofs," "Can't Computers Solve Everything? .... Imposs ib le Prob- lems." Every sen tence is clear and com- prehens ib le . The love of mathemat ics that impels Stewart is a lways there; if the r eade r is suscept ib le at all, she or he may well b e c o m e infected.

Starting with Chapter 14, and going on to Chapter 20, the next to last, there is a not iceable change of tone and focus. "The Career Ladder," "Pure or Applied," "Where Do You Get Those Crazy Ideas?" "How to Teach Math," "The Mathemati- cal Community," "Perils and Pleasures of Collaboration." Ian is no longer talking to a child, sharing his enthusiasm and en- joyment. He is talking to someone who is commit ted to mathematics, and is wor- r ied about h o w to make a living at it.

Of course , this is a very realistic k ind of conversa t ion to imagine. In fact, many senior mathematicians , responsi- ble for gu id ing advanced undergradu- ates, g radua te students, postdocs , and faculty just s tart ing to teach, do have such conversa t ions many t imes over their t each ing careers. I imagine that whi le the "Meg" of the first half of the b o o k is par t ly b a s e d on real acquain- tance with schoo l children, and partly a creat ion of Stewart 's imagination, the "Meg" of the s e c o n d part may well be an a ma lga m of many y o u n g mathe- maticians Stewart has counseled.

So wha t k ind of advice does he give? I w o u l d say, s o u n d and sober advice. Realist advice, h o w to get on in the math- ematical wor ld as it really is. (Meaning, of course, not necessari ly as w e would most, in our hear t of hearts, desire it to be.) Stewart k n o w s what 's what, and he most k indly and sincerely wants Meg to make it, to get that job and that tenure. That means , k n o w i n g what hir ing com- mit tees l ook for, and wha t p romo- t ion and t enure commit tees look for, etc.etc.etc. "THE REAL WORLD."

So, very good , wha t could be wrong with that?

Nothing at all. And yet I can ' t he lp r emember ing a

THE MATHEMATICAL INTELLIGENCER